Asymptotic stability for the Couette flow in the 2D Euler equations

TL;DR

This paper discusses the nonlinear asymptotic stability of the Couette flow in the 2D Euler equations, showing that small perturbations converge to a nearby shear flow and enstrophy is transferred to small scales over time.

Contribution

It provides a detailed exposition of recent results proving the nonlinear stability of Couette flow in 2D Euler equations with insights into the proof techniques.

Findings

Perturbations to Couette flow converge strongly in L^2.

Enstrophy is transferred to small scales and lost in the weak limit.

The stability holds for perturbations small in a suitable regularity class.

Abstract

In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is proved that perturbations to the Couette flow which are small in a suitable regularity class converge strongly in $L^2$ to a shear flow which is close to the Couette flow. Enstrophy is mixed to small scales by an almost linear evolution and is generally lost in the weak limit as t -> +/- infinity. In this note we discuss the most important physical and mathematical aspects of the result and the key ideas of the proof.